In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/10 + i sin π/10)]⁵

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Recall DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)th power is given by \(r^n (\cos(n\theta) + i \sin(n\theta))\).

Identify the given complex number's modulus and argument: here, the modulus \(r = \frac{1}{2}\) and the argument \(\theta = \frac{\pi}{10}\).

Apply DeMoivre's Theorem to raise the complex number to the 5th power: compute \(r^5\) and multiply the argument by 5, so the expression becomes \(\left(\frac{1}{2}\right)^5 \left( \cos \left(5 \times \frac{\pi}{10} \right) + i \sin \left(5 \times \frac{\pi}{10} \right) \right)\).

Simplify the argument inside the trigonometric functions: \(5 \times \frac{\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2}\).

Write the final expression in rectangular form by evaluating \(\cos \frac{\pi}{2}\) and \(\sin \frac{\pi}{2}\), then multiply each by \(\left(\frac{1}{2}\right)^5\) to get the real and imaginary parts.

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주요 개념

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DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It allows raising complex numbers to integer powers by multiplying the angle and raising the magnitude to the power, simplifying calculations.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting between these forms is essential, especially after applying DeMoivre's Theorem, to write the final answer in the requested rectangular form.

Trigonometric Identities for Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by n. Understanding trigonometric identities and how to evaluate cos(nθ) and sin(nθ) accurately is crucial for simplifying the expression and converting it back to rectangular form.